# 0.4 Time-frequency dictionaries  (Page 2/3)

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It can be interpreted as a Fourier transform of $\phantom{\rule{0.166667em}{0ex}}f$ at the frequency ξ , localized by the window $g\left(t-u\right)$ in the neighborhood of u . This windowed Fourier transform is highly redundant and represents one-dimensional signalsby a time-frequency image in $\left(u,\xi \right)$ . It is thus necessary to understand how to select many fewertime-frequency coefficients that represent the signal efficiently.

When listening to music, we perceive sounds that have a frequency that varies in time. Chapter 4 showsthat a spectral line of $\phantom{\rule{0.166667em}{0ex}}f$ creates high-amplitude windowed Fourier coefficients $Sf\left(u,\xi \right)$ at frequencies $\xi \left(u\right)$ that depend on time u . These spectral components are detected and characterized byridge points, which are local maxima in this time-frequency plane. Ridge points define a time-frequency approximation support λ of $\phantom{\rule{0.166667em}{0ex}}f$ with a geometry that depends on the time-frequency evolution of the signal spectral components. Modifying thesound duration or audio transpositions are implemented by modifying the geometry of the ridge support in time frequency.

A windowed Fourier transform decomposes signals over waveforms that have the same time and frequency resolution. It is thus effectiveas long as the signal does not include structures having different time-frequency resolutions, some being very localizedin time and others very localized in frequency.  Wavelets address this issue by changing the time and frequency resolution.

## Continuous wavelet transform

In reflection seismology, Morlet knew that the waveforms sent underground have a duration that is too longat high frequencies to separate the returns of fine, closely spaced geophysical layers. Such waveforms are called wavelets in geophysics. Instead of emitting pulses of equal duration,he thought of sending shorter waveforms at high frequencies. These waveforms were obtained by scaling the motherwavelet, hence the name of this transform. Although Grossmann was working in theoretical physics, he recognized in Morlet's approach some ideasthat were close to his own work on coherent quantum states.

Nearly forty years after Gabor, Morlet and Grossmann reactivated a fundamentalcollaboration between theoretical physics and signal processing, whichled to the formalization of the continuous wavelet transform(GrossmannM:84). These ideas were not totally new to mathematicians working in harmonic analysis, or to computer visionresearchers studying multiscale image processing. It was thus only the beginning of a rapid catalysis that brought togetherscientists with very different backgrounds.

A wavelet dictionary is constructed from a mother wavelet ψ of zero average

${\int }_{-\infty }^{+\infty }\psi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=0,$

which is dilated with a scale parameter s , and translated by u :

$\begin{array}{c}\hfill \mathcal{D}={\left\{{\psi }_{u,s},\left(t\right),=,\frac{1}{\sqrt{s}},\phantom{\rule{0.166667em}{0ex}},\psi ,\left(\frac{t-u}{s}\right)\right\}}_{u\in \mathbb{R},s>0}.\end{array}$

The continuous wavelet transform of $\phantom{\rule{0.166667em}{0ex}}f$ at any scale s and position u is the projection of $\phantom{\rule{0.166667em}{0ex}}f$ on the corresponding wavelet atom:

$\begin{array}{c}\hfill W\phantom{\rule{0.166667em}{0ex}}f\left(u,s\right)=⟨\phantom{\rule{0.166667em}{0ex}}f,{\psi }_{u,s}⟩={\int }_{-\infty }^{+\infty }f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\frac{1}{\sqrt{s}}\phantom{\rule{0.166667em}{0ex}}{\psi }^{*}\left(\frac{t-u}{s}\right)\phantom{\rule{-0.166667em}{0ex}}dt.\end{array}$

It represents one-dimensional signals by highly redundant time-scale images in $\left(u,s\right)$ .

## Varying time-frequency resolution

As opposed to windowed Fourier atoms, wavelets have a time-frequency resolution that changes.The wavelet ${\psi }_{u,s}$ has a time support centered at u and proportional to s . Let us choose a wavelet ψ whose Fourier transform $\stackrel{^}{\psi }\left(\omega \right)$ is nonzero in a positive frequency interval centered at η . The Fourier transform ${\stackrel{^}{\psi }}_{u,s}\left(\omega \right)$ is dilated by $1/s$ and thus is localized in a positive frequencyinterval centered at $\xi =\eta /s$ ; its size is scaled by $1/s$ . In the time-frequency plane, the Heisenberg boxof a wavelet atom ${\psi }_{u,s}$ is therefore a rectangle centered at $\left(u,\eta /s\right)$ , with time and frequency widths, respectively,proportional to s and $1/s$ . When s varies, the time and frequency width of this time-frequency resolution cell changes, butits area remains constant, as illustrated by [link] .

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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Berger describes sociologists as concerned with
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